On the Convex Pfaff-Darboux Theorem of Ekeland and Nirenberg

The classical Pfaff-Darboux theorem, which provides local 'normal forms' for $1$-forms on manifolds, has applications in the theory of certain economic models [Chiappori P.-A., Ekeland I., Found. Trends Microecon. 5 (2009), 1-151]. However, the normal forms needed in these models often come with an additional requirement of some type of convexity, which is not provided by the classical proofs of the Pfaff-Darboux theorem. (The appropriate notion of 'convexity' is a feature of the economic model. In the simplest case, when the economic model is formulated in a domain in $\mathbb{R}^n$, convexity has its usual meaning.) In [Methods Appl. Anal. 9 (2002), 329-344], Ekeland and Nirenberg were able to characterize necessary and sufficient conditions for a given 1-form $\omega$ to admit a convex local normal form (and to show that some earlier attempts [Chiappori P.-A., Ekeland I., Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4 25 (1997), 287-297] and [Zakalyukin V.M., C. R. Acad. Sci. Paris S\'er. I Math. 327 (1998), 633-638] at this characterization had been unsuccessful). In this article, after providing some necessary background, I prove a strengthened and generalized convex Pfaff-Darboux theorem, one that covers the case of a Legendrian foliation in which the notion of convexity is defined in terms of a torsion-free affine connection on the underlying manifold. (The main result of Ekeland and Nirenberg concerns the case in which the affine connection is flat.)


Introduction
The Pfaff-Darboux theorem provides a local 'normal form' for 1-forms on manifolds, assuming that certain constant rank conditions are met.A common version 1 of this classical theorem is the following: Let ω be a smooth 1-form on an n-manifold M and suppose that there is an integer k > 0 such that ω ∧ (dω) k vanishes identically on M while ω ∧ (dω) k−1 is nowhere vanishing on M .
Then each m ∈ M has an open neighborhood U ⊂ M on which there exist (smooth) functions y 1 , . . ., y k , p 2 , . . ., p k and a nonvanishing function a such that2 U * ω = a(dy 1 + p 2 dy 2 + • • • + p k dy k ). (1.1) Since the functions y 1 , . . ., y k , p 2 , . . ., p k in this representation must be independent on U .The normal form (1.1) is often written more symmetrically as where the a i do not simultaneously vanish in U .In this representation, the independence of the functions y 1 , . . ., y k , p 2 , . . ., p k translates into the condition that the mapping In fact, the representation (1.2) is more common in treatises on mathematical economics, where the Pfaff-Darboux theorem plays an important role [3].Often, the normal forms needed in these models sometimes come with an additional requirement of convexity, i.e., the underlying manifold is M = R n (or an open domain in R n ), and one would like to arrange that the functions a i be positive and the functions u i be strictly convex, i.e., have positive definite Hessians. 3 useful reference for the role of convexity in economic models is the book [4] and the article [1].Now, it turns out that constructing such a convex Pfaff-Darboux representation is not always possible, which raises the question of how to determine when one exists.In [5], Ekeland and Nirenberg were able to provide necessary and sufficient conditions for a given 1-form ω ∈ Ω 1 (R n ) to admit a local convex Pfaff-Darboux normal form.They also constructed examples that showed that some earlier attempts [3,6] to find such conditions had been unsuccessful.
In this note, after providing some necessary background, I prove a generalization of the convex Pfaff-Darboux theorem of Ekeland and Nirenberg.This treatment has some notable features that make it of interest for the general problem.
First, the proof of Ekeland and Nirenberg does not assume the classical Pfaff-Darboux theorem; instead, it constructs the desired convex representation directly using the Frobenius theorem, essentially reproving the Pfaff-Darboux theorem but with the additional convexity condition imposed.The proof in this article assumes the classical Pfaff-Darboux theorem, so that the argument can more directly focus on choosing a Pfaff-Darboux representation that satisfies the convexity requirements.This results in a shorter proof, one that also brings the nature of the convexity requirements more sharply into focus.
Second, the notion of strict convexity turns out to be meaningful on any manifold endowed with a torsion-free affine connection, and the proof below covers this more general situation with no extra work.
Third, the proof yields a stronger result, in that it produces a local convex Pfaff-Darboux representation of ω adapted to any ω-Legendrian foliation that satisfies a certain geometrically natural positivity condition, one that is equivalent pointwise to the condition of Ekeland and Nirenberg.

Canonical subbundles
An ω satisfying (2.1) and (2.2) defines a kernel subbundle K = ω −1 (0) ⊂ T M of corank 1 and a subbundle A ⊂ K of corank 2(k−1) in K by the rule that, for each m ∈ M , Replacing ω by ω = f ω for any nonvanishing function f does not change K or A.
If k > 1, then K ⊂ T M is not an integrable plane field, but the subbundle A ⊂ T M is always integrable, since it is the Cauchy characteristic plane field of the differential ideal I generated by ω (see [2, Chapter II, Proposition 2.1]).In the contact case, i.e., when n = 2k−1 (which is, in some sense, generic), one has A = (0).
There is a nondegenerate, skew-symmetric bilinear pairing B ω : when v, w ∈ K m .It satisfies B f ω = f B ω for any nonvanishing function f on M .Note that any subspace W ⊂ T m on which both ω and dω vanish, must, first of all, satisfy W ⊂ K m (since ω vanishes on W ), and, second, must have codimension at least k−1 in K m , since dω, as a skew-symmetric form on K m , has Pfaff rank k−1.Moreover, if W does have codimension k−1 in K m , then it must contain A m , so that W/A m is a null subspace of B ω .

Legendrian submanifolds and Grassmannians
Any submanifold L ⊂ M that satisfies L * ω = 0, i.e., an integral manifold of ω, must also satisfy L * dω = 0 and hence, by the above linear algebra discussion, must have codimension at least k in M .When L ⊂ M is an integral manifold of ω of codimension k, it is said to be an ω-Legendrian submanifold.
In particular, if This motivates defining the Legendrian Grassmannian Leg m (ω) ⊂ Gr k (T m M ) to be the set of subspaces W ⊂ K m that have codimension k in T m M and on which both ω and dω vanish.By the above remarks, it follows that Leg m (ω) can be canonically identified with the is a smooth subbundle, and Leg(f ω) = Leg(ω) for all nonvanishing f .

A local normal form
One version of the Pfaff-Darboux theorem [2, Chapter II, Theorem 3.1] states that, when ω ∈ Ω 1 (M ) satisfies (2.1) and (2.2), each m ∈ M has an open neighborhood U ⊂ M on which there exist smooth functions u 1 , . . ., u k and a 1 , . . ., a k (with not all a i simultaneously vanishing) so that Moreover, the mapping (u, [a]) : U → R k ×RP k−1 is a submersion.(In fact, the kernel subbundle of the differential of this mapping is the restriction of A to U .) Conversely, the existence of functions u i and a i for 1 ≤ i ≤ k on an open set U ⊂ M satisfying (2.3) with the a i not all simultaneously vanishing and having the property that (u, [a]) : U → R k × RP k−1 be a submersion implies that both (2.1) and (2.2) hold on U .

Geometry of the normal form
It will be useful to have a geometric interpretation of the Pfaff-Darboux theorem.Now, in the representation (2.3), the functions u i have independent differentials, i.e., du 1 ∧ • • • ∧ du k does not vanish on U .Consequently, the simultaneous level sets of the functions u i define a foliation L of U ⊂ M by ω-Legendrian submanifolds, i.e., an ω-Legendrian foliation.
Conversely, given an ω-Legendrian foliation L on an open subset V ⊂ M , each point m ∈ V will have an open neighborhood U ⊂ V in which the leaves of L are the fibers of a submersion u = (u i ) : U → R k .Since ω vanishes when pulled back to any fiber of u, it follows that there exists a mapping a = (a i ) : Thus, a geometric interpretation of the Pfaff-Darboux theorem is the statement that, when ω ∈ Ω 1 (M ) satisfies (2.1) and (2.2), each point m ∈ M has an open neighborhood U ⊂ M on which there exists an ω-Legendrian foliation.

Variants and extensions
There are a number of variants and extensions of the classical Pfaff-Darboux theorem that can all be seen to be equivalent to the above versions by elementary arguments [2, Chapter II, Section 3].In this article, two such variants will be important.For convenience of reference, they will be stated as propositions.3 Convexity and affine manifolds

Classical convexity
When M = R n , there is a notion of strict convexity of a function u, which is the condition that the Hessian quadratic form H(u) be positive definite, where and where x 1 , . . ., x n are the usual affine linear coordinates in R n .Note that strict convexity is an affine-invariant notion on R n .Motivated by applications in economics, Ekeland and Nirenberg [5] asked what further conditions one must impose on an ω ∈ Ω 1 (R n ) satisfying (2.1) and (2.2) in order to know that one can choose the functions u j and a j in the representation (2.3) so that the u j be strictly convex and the a j be positive.It is not hard to show, by example, that some further condition on ω is necessary to guarantee the existence of such a convex representation.(See the discussion at the beginning of Section 3.3.) They showed that two earlier articles [3,6] claiming to provide such necessary and sufficient conditions were flawed (indeed, they exhibited counterexamples to the claims of these articles) and then produced their own condition, which they showed to be necessary and sufficient.
In this note, I will show that their main result, properly formulated, holds good on an nmanifold M endowed with a torsion-free affine connection, not just on R n endowed with the (flat) affine connection it inherits as a vector space.

Affine connections and convexity
Let ∇ be a torsion-free affine connection on an n-manifold M n , i.e., ∇ is a first-order, linear differential operator for all smooth functions f on M and smooth 1-forms η on M .The assumption that ∇ be torsion-free is the condition that the associated (second-order) Hessian operator H(u) = ∇(du) be a symmetric (0, 2)-tensor for each smooth function u on M .
A smooth function u on M is said to be strictly ∇-convex if, as a quadratic form, H(u) is positive definite at every point of M .
When M = R n and ∇ is the standard (flat) connection, satisfying ∇(dx i ) = 0 for all of the coordinate functions x i , then H(u) is the usual Hessian tensor (3.1), and this notion of convexity is simply the classical one.
In the more general case, when x = (x i ) : U → R n is a local coordinate chart, one has are the coefficients of the connection ∇ relative to the coordinate chart x = (x i ).The general coordinate formula for H then becomes Thus, ∇-convexity of u is expressible in terms of a condition on the 2-jet of u, slightly more general than the condition for classical convexity.
Adopting the usual conventions one sees that, for a 1-form ω of the form one has (using the summation convention) where I have introduced the notation S ∇ ω to denote the symmetrization of ∇(ω).Thus, S ∇ ω = ∇ω − dω is a well-defined quadratic form on M .(Of course, the linear, first-order differential operator S ∇ depends on ∇.)

A positivity condition
The quadratic form S ∇ ω provides some insight into the question of whether a ∇-convex Pfaff-Darboux representation of ω is possible.
Proposition 3.1.Suppose that ω ∈ Ω 1 (M ) satisfies (2.1) and (2.2).If there exist positive functions a i and ∇-convex functions u i for 1 ≤ i ≤ k such that (3.2) holds, then S ∇ ω is positive definite on the leaves of the foliation L defined by Proof .Since S ∇ ω = da i • du i + a i H(u i ), it follows that, when restricted to the plane field N ⊂ T M defined by du 1 = du 2 = • • • = du k = 0, the terms da i • du i in S ∇ ω vanish.Thus, S ∇ ω = a i H(u i ) as quadratic forms on N .By the positivity of the a i and the ∇-convexity of the u i , it follows that S ∇ ω is positive definite on N . 4 ■ This proposition provides necessary condition for the existence of a convex Pfaff-Darboux representation.
Example 3.2 (an obstructed example).Let M = R n with standard coordinates x = (x i ), and let ∇ be the (flat, torsion-free) connection such that ∇(dx i ) = 0 for 1 ≤ i ≤ n.Let c i , f ij = −f ji , and g ij = g ji be constants and consider the 1-form Now assume that the skew-symmetric matrix f = (f ij ) has rank 2(k−1) < n, so that (dω) k−1 ̸ = 0 but (dω) k = 0.Then, for generic choice of the constants c i , ω ∧ (dω) k−1 will be nonvanishing on an open neighborhood U ⊂ R n of the origin 0 ∈ R n , in which case, ω will satisfy the hypotheses (2.1) and (2.2) on U .
If the symmetric matrix g = (g ij ) does not have at least n−k positive eigenvalues, then S ∇ ω cannot be positive definite on any (n−k)-dimensional subbundle N ⊂ T U , and, hence, by Proposition 3.1, ω cannot have a convex Pfaff-Darboux representation in any open subset of U .
It turns out that this necessary condition for a local 'convex' Pfaff-Darboux representation compatible with an ω-Legendrian foliation L is also sufficient.Theorem 3.3.Suppose ∇ be a torsion-free affine connection on M , that ω ∈ Ω 1 (M ) satisfy (2.1) and (2.2) for some k > 0, and that L be an ω-Legendrian foliation on M with the property that S ∇ ω pulls back to each leaf of L to be positive definite.Then each m ∈ M has an open neighborhood U ⊂ M on which there exist strictly ∇-convex functions u 1 , . . ., u k that are constant on the leaves of L in U and positive functions a 1 , . . ., a k such that 4 It is worth pointing out that the same conclusion about the positive definiteness of S ∇ ω on the leaves of L would have followed if one had merely assumed that each u i be only 'strictly ∇-quasi-convex', i.e., that du i be nonvanishing and H(u i ) be positive definite when restricted to the hyperplane field du i = 0. Compare [5, Lemma 1], and the preceding discussion about their Problem 2.
Before giving the proof of Theorem 3.3, I will state one of its corollaries, so that it can be compared with the main result of Ekeland and Nirenberg [5, Theorem 1].
First, some useful terminology.As always, assume that ω satisfies (2.1) and (2.2) for some k > 0. Definition 3.4.An ω-Legendrian subspace W ⊂ T m M is ∇-positive for ω if the restriction of the quadratic form S ∇ ω to W is positive definite.
Let Leg + (ω, ∇) ⊆ Leg(ω) denote the set of ω-Legendrian subspaces that are ∇-positive for ω.Then Leg + (ω, ∇) is a (possibly empty) open subset of Leg(ω).Consequently, the set of points m ∈ M for which there exists a ∇-positive, ω-Legendrian subspace W ⊂ T m M is an open subset of M .Also, note that, since such a W contains A m , it follows that S ∇ ω must be positive definite on A m .Corollary 3.5.Suppose that ∇ be a torsion-free affine connection on M , that ω ∈ Ω 1 (M ) satisfy (2.1) and (2.2) for some k > 0, and that there exist a W ∈ Leg + (ω, ∇) with W ⊂ T m M .Then m ∈ M has an open neighborhood U ⊂ M on which there exist strictly ∇-convex functions u 1 , . . ., u k and positive functions a 1 , . . ., a k such that The proof of Corollary 3.5 follows by applying Propositions 2.1 and 2.2 to produce an ω-Legendrian foliation L on an open neighborhood V of m whose leaf through m has W as a tangent space.Since S ∇ ω is positive definite on W , it follows that it is positive definite on all the tangent spaces to the leaves of L in some (possibly) smaller m-neighborhood V ′ ⊂ V .Now apply Theorem 3.3 to L on V ′ .Remark 3.6.In the special case in which M = R n and ∇ is the flat connection satisfying ∇(dx i ) = 0 for x i the standard coordinates on R n , Corollary 3.5 simply becomes Theorem 1 of Ekeland and Nirenberg [5], since their Condition 3 turns out to be equivalent to the existence of a W ∈ Leg + (ω, ∇) with W ⊂ T m M in this case.
Proof of Theorem 3.3.There exists an m-neighborhood V 0 ⊂ M on which there exist smooth functions y 1 , . . ., y k vanishing at m so that the leaves of dy 1 = • • • = dy k = 0 are intersections of the leaves of L with V 0 as well as functions p 2 , . . ., p k , also vanishing at m, and a nonvanishing function a so that By reversing the signs of a and the y i , if necessary, one can assume that a(m) > 0. Let W ⊂ T m M be the tangent to the leaf of L that passes through m, so that W is the common kernel of the dy i evaluated at m. Set ω = a −1 ω and note that, since dω ≡ a −1 dω mod ω, it follows that L is also ω-Legendrian.Moreover, since it follows that the tangent spaces of L (which, of course, satisfy ω = 0) are also ∇-positive for ω.Since ω = aω and a > 0, finding the desired convex representation for ω will also yield one for ω.Thus, it suffices to prove the theorem with ω in the place of ω, i.e., to assume that a = 1, so I will do that from now on.Thus, Since ω ∧ (dω) k−1 ̸ = 0, the functions y 1 , . . ., y k , p 2 , . . ., p k have linearly independent differentials at m. Restricting to V 0 , i.e., setting M = V 0 , one has Since the p j vanish at m, it follows that, when restricted to W ⊂ T m M , the two quadratic forms H y 1 and S ∇ ω are equal.Thus H y 1 is positive definite on W , and so there is a constant c > 0 so that H y 1 + c dy Thus, one could have chosen the functions y 1 , . . ., y k , p 2 , . . ., p k with H y 1 being positive definite on the hyperplane K m .Assume now that this was done.
It still needs to be shown that one can choose the functions y 1 , . . ., y k , p 2 , . . ., p k with H y 1 being positive definite on all of T m M , not just on K m , which is the kernel of dy 1 at m.To do this, note that, if ϕ is any smooth function on a neighborhood of the origin in R, then H ϕ y 1 = ϕ ′ y 1 H y 1 + ϕ ′′ y 1 dy 1 2 .

Proposition 2 . 1 .Proposition 2 . 2 .
Suppose that ω ∈ Ω 1 (M ) satisfies (2.1) and (2.2).Then for each m ∈ M and W ∈ Leg m (ω), there exists a ω-Legendrian submanifold L ⊂ M such that m ∈ L and W = T m L. Suppose that ω ∈ Ω 1 (M ) satisfies (2.1) and (2.2) and that L ⊂ M is an embedded ω-Legendrian submanifold.Then each m ∈ L has an open neighborhood U ⊂ M on which there exists an ω-Legendrian foliation L with the property that L ∩ U is a leaf of L.